Undecidable Problems for Probabilistic Automata of Fixed Dimension

Abstract. We prove that several problems associated with probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matrices with nonnegative rational entries is bounded is undecidable.     

Optimal Finite Characterization of Linear Problems with Inexact Data

Abstract. For many linear problems, in order to check whether a certain property is true for all matrices A from an interval matrix A, it is sufficient to check this property for finitely many “vertex” matrices A ∈ A. J. Rohn has discovered that we do not need to use all 2ⁿ² vertex matrices, it is sufficient to only check these properties for 2²ⁿ⁻¹ ≪ 2ⁿ² vertex matrices of a special type A_yz. In this paper, we show that a further reduction is impossible: without checking all 2²ⁿ⁻¹ matrices A_yz, we cannot guarantee that the desired property holds for all A ϵ A. Thus, these special vertex matrices provide an optimal finite characterization of linear problems with inexact data.     

微结构网点的半色调信息隐藏技术

目的 基于阈值矩阵的加网方法设计出具有微观防伪功能的艺术型网点,利用半色调信息隐藏技术将隐藏信息载入到由微结构网点组成的色版中,研究微结构网点在信息隐藏中的应用价值。方法 利用改进后的阈值矩阵对要加载隐藏信息的色版进行微结构加网,完成各色版的半色调处理,将隐藏信息嵌入到微结构加网的色版中,利用参数匹配的解锁工具完成信息的提取,并评价信息的隐藏和提取效果。结果 通过信息的隐藏和提取实验,主观测试发现无论是单色版还是合并色版,微结构网点对信息隐藏的视觉保密性效果都比普通网点的隐蔽性好,客观结构相似度的数值计算结果都达到了0.995以上,比普通隐藏结果更佳,而且显隐能够达到信息清晰完整的再现效果。结论 该防伪方法能够在不增加成本来的条件下提高经济价值较高的印刷品和包装品的防伪标准、效率和附加价值,尤其在有价证券、护照、票据等印刷品、医药食品包装和电子影像出版产品的包装防伪方面具有广阔的应用前景与价值。     

Equivalence transformations of rational matrices and applications

The known theories of transformations between polynomial matrices are extended to the case of rational matrices. Specifically, Ω-equivalence between rational matrices having possibly different dimensions is defined, and this has the property of preserving the zero structure of rational matrices in the region Ω ? C ∪ {∞}Some implications of this new equivalence transformation for linear system theory are also provided.     

Eigenvalue assignment in linear multivariable systems by output feedback

The assignment of arbitrary cloacd-loop eigenvalue spectra by output feedback is investigated in the special ease of systems for which the set of state-feedback matrices P which yield a desired closed-loop eigenvalue spectrum is finite and can be readily generated. In particular, a simple necessary and sufficient condition for the existence of an output-feedback matrix K corresponding to a prescribed state-feedback matrix P is established which also leads directly to the matrix K in cases when the condition is satisfied. The condition derived in this paper requires neither the computation of generalized inverse matrices (Munro and Vardulakis 1973), nor the transformation of state vectors (Patel 1974), nor the determination of Luenberger canonical forms (Vardulakis 1976), and indicates very clearly that an output-feedback matrix K corresponding to a prescribed state-feedback matrix P exists only in very special circumstances.     

Convergence and comparison results for double splittings of Hermitian positive definite matrices

Abstract For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10     

Piecewise separable matrices

Abstract Piecewise separable matrices are introduced as a natural generalization of semiseparable and profile-low-rank matrices. We show that such matrices possess linear matrix-by-vector complexity.     

Mixed and componentwise condition numbers for rectangular structured matrices

Abstract  We provide explicit expressions for both mixed and componentwise structured condition numbers for several classes of structured rectangular matrices: upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. Mathematics Subject Classification (2000): 15A12, 65F35     

On general ℋ2 control: from frequency to time domain

This article presents a general formula for discrete-time ?₂ control. It works with the regular and singular case of ?₂ control, i.e. in the case of possibly non-left-invertible matrices G ₁₂ and non-right-invertible matrices G ₂₁, with possible unit circle invariant zeros. In the generic case, it can be simplified and adapted to work with the plant transfer matrix directly, without invoking the matrices of the parameterisation of stabilising controllers. A further result of this article is the presented necessary and sufficient conditions for state-space ?₂ control, under only stabilisability and detectability assumptions. If the conditions are satisfied, an observer-based ?₂ controller is constructed. The corresponding numerical algorithm consists of solving two discrete-time algebraic Riccati systems (DARSs) and two eigen-problems.     

SYMMLQ-like procedure for Ax = b where A is a special normal matrix

Abstract We propose a method of Lanczos type for solving a linear system with a normal matrix whose spectrum is contained in a second-degree curve. This is a broader class of matrices than that of the (l,m)-normal matrices introduced in a recent paper by Barth and Manteuffel. Our approach is similar to that of Huhtanen in the sense that both use the condensed form of normal matrices discovered by Elsner and Ikramov. However, there are a number of differences, among which are: (i) our method is modeled after the SYMMLQ algorithm of Paige and Saunders; (ii) it uses only one matrix-vector product per step; (iii) we provide effective means for monitoring the size of the residual during the process. Numerical experiments are presented. Keywords: generalized Krylov sequence, Rayleigh-Ritz projection, SYMMLQ algorithm, GMRES algorithm, normal matrix     

ARMA Model order estimation using third order computations

A novel approach for estimating the order of a non-Gaussian autoregressive moving-average process is proposed. The proposed method uses third order computations, and is based on the minimum eigenvalue of a family of covariance matrices derived from the observed output data. The algorithm uses data matrices rather than calculated cumulants. Hence, we avoid the non-stationary effects, which is due to finite-length observations. Examples are given to demonstrate the performance of the proposed algorithm.     

Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

In this paper, we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the preconditioned conjugate gradient solution of the linearized Newton system to solve Au=q(u) u, q(u) being the Rayleigh quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi–Davidson method on all the test problems.     

A New Upper Bound For Moduli Of Eigenvalues Of Iterative Matrices

In this paper, we present a new upper bound for the spectral radius of iterative matrices based on the concept of doubly diagonal dominance, which improved the result of paper [1]. As its applications, for example, we apply it to the AOR and MSOR iterative matrices.     

Quantum Algorithms for Weighing Matrices and Quadratic Residues

Abstract. In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to devise new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein and Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol χ (which indicates if an element of a finite field F _q is a quadratic residue or not). It is shown how for a shifted Legendre function f _s (i)=χ(i+s) , the unknown s ∈ F _q can be obtained exactly with only two quantum calls to f _s . This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log q + log ((1-ɛ )/2) queries to solve the same problem.     

On the choice of dimensionality and sample size for feature selection

Abstract

This paper considers the problem of selection of dimensionality and sample size for feature extraction in pattern recognition. In general, the axes of the feature space are selected as the eigenvectors of matrices of the form R ₂ ^?1 R ₁ where R ₁ and R ₂ are real symmetric matrices. Expressions are derived for obtaining the changes in the eigenvalues and eigenvectors when there are changes of first order of smallness in the matrices R ₁ and R ₂. Based on this theory, a method is developed for choosing the dimensionality of the patterns. Also expressions are derived for the selection of sample size for estimating the eigenvectors, for two gaussian distributed pattern classes with equal means, unequal covariance matrices and with unequal means and equal covariance matrices.     

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .     

Convex invertible cones of state space systems

In the paper [CL1] the notion of a convex invertible cone,cic, of matrices was introduced and its geometry was studied. In that paper close connections were drawn between thiscic structure and the algebraic Lyapunov equation. In the present paper the same geometry is extended to triples of matrices andcics of minimal state space models are defined and explored. This structure is then used to study balancing, Hankel singular values, and simultaneous model order reduction for a set of systems. State spacecics are also examined in the context of the so-called matrix sign function algorithm commonly used to solve the algebraic Lyapunov and Riccati equations.     

Basic Lul Factorization And Improved Iterative Method With Orderings

Consider the 'basic LUL factorization' of the matrices as the generalization of the LU factorization and the UL factorization, and using this LUL factorization of the matrices, we propose an "improved iterative method" such that the spectral radius of this iterative matrix is equal to zero, and this method converges at most n iterations. Our main concern is the necessary and sufficient conditions that the improved iterative matrix is equal to the iterative matrix of the improved SOR method with orderings. Concerning the tridiagonal matrices and the upper Hessenberg matrices, this method becomes the improved SOR method with orderings, and we give n selections of the multiple relaxation parameters such that the spectral radii of the corresponding improved SOR matrices are 0. We extend these results to a class of $n \times n$ matrices. We also consider the basic LUL factorization and improved iterated method 'corresponding to permutation matrices'.     

Sufficient conditions for asymptotic Liapunov stability of special system of linear differential equations with constant coefficients

The sufficient conditions for asymptotic Liapunov stability of block 2 n 2 2 n matrices, which consist of four n 2 n matrices, are presented. These conditions are applied to the special system arising in connection with optimization problems for gas transportation and distribution.     

Simple criteria for generalized diagonally dominant matrices

Abstract

In this paper we provide several new criteria for generalized diagonally dominant matrices (GDDMs) by making use of elements of matrices only, and also propose two corresponding non-parameter algorithms to test GDDMs. Numerical examples for the effectiveness of the methods are presented.